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1: 28.17 Stability as
§28.17 Stability as
… ►For real and the stable regions are the open regions indicated in color in Figure 28.17.1. The boundary of each region comprises the characteristic curves and ; compare Figure 28.2.1. … ►2: 12.20 Approximations
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►Luke (1969b, pp. 25 and 35) gives Chebyshev-series expansions for the confluent hypergeometric functions and (§13.2(i)) whose regions of validity include intervals with endpoints and , respectively.
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3: 10.72 Mathematical Applications
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►In regions in which (10.72.1) has a simple turning point , that is, and are analytic (or with weaker conditions if is a real variable) and is a simple zero of , asymptotic expansions of the solutions for large can be constructed in terms of Airy functions or equivalently Bessel functions or modified Bessel functions of order (§9.6(i)).
These expansions are uniform with respect to , including the turning point and its neighborhood, and the region of validity often includes cut neighborhoods (§1.10(vi)) of other singularities of the differential equation, especially irregular singularities.
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►In regions in which the function has a simple pole at and is analytic at (the case in §10.72(i)), asymptotic expansions of the solutions of (10.72.1) for large can be constructed in terms of Bessel functions and modified Bessel functions of order , where is the limiting value of as .
These asymptotic expansions are uniform with respect to , including cut neighborhoods of , and again the region of uniformity often includes cut neighborhoods of other singularities of the differential equation.
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4: 21.10 Methods of Computation
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5: Bonita V. Saunders
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►This work has resulted in several published papers presented as contributed or invited talks at universities and regional, national, and international conferences.
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6: 5.10 Continued Fractions
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5.10.1
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7: 14.31 Other Applications
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►The conical functions appear in boundary-value problems for the Laplace equation in toroidal coordinates (§14.19(i)) for regions bounded by cones, by two intersecting spheres, or by one or two confocal hyperboloids of revolution (Kölbig (1981)).
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8: 33.23 Methods of Computation
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►Bardin et al. (1972) describes ten different methods for the calculation of and , valid in different regions of the ()-plane.
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►Hull and Breit (1959) and Barnett (1981b) give WKBJ approximations for and in the region inside the turning point: .
9: 19.7 Connection Formulas
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►The first of the three relations maps each circular region onto itself and each hyperbolic region onto the other; in particular, it gives the Cauchy principal value of when (see (19.6.5) for the complete case).
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►The second relation maps each hyperbolic region onto itself and each circular region onto the other:
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►The third relation (missing from the literature of Legendre’s integrals) maps each circular region onto the other and each hyperbolic region onto the other:
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10: 19.21 Connection Formulas
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►Change-of-parameter relations can be used to shift the parameter of from either circular region to the other, or from either hyperbolic region to the other (§19.20(iii)).
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►For each value of , permutation of produces three values of , one of which lies in the same region as and two lie in the other region of the same type.
In (19.21.12), if is the largest (smallest) of , and , then and lie in the same region if it is circular (hyperbolic); otherwise and lie in different regions, both circular or both hyperbolic.
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